Inhaltsbereich

Branching Processes Lingering around their Carrying Capacity

Zusammenfassung:
Independence of reproducing individuals can be viewed as the
very defining property of branching processes. It is crucial for the most
famous results of the theory, the determination of the extinction probability
and the dichotomy between extinction and exponential increase. We introduce
general branching processes in a habitat with a finite carrying capacity, so
that independence between individuals is destroyed. Indeed, our populations
are supercritical below and subcritical above the carrying capacity. We also
show that small such populations may reach the carrying capacity, and once
there they will linger around it for a time that is exponential in the
carrying capacity.